Prime Factorisation

Learning Objective:

• we are learning to write an integer as a product of prime numbers.

Investing:

• This is useful because the prime factors of a number decide many of the properties of a number and how it behaves with other numbers.
  
  
• A functional (real-life) application is public key encryption over the internet. The RSA algorithm and other similar methods involve multiplying two very large prime numbers to make a huge number that is very hard to factorize. This can be used to securely encode your emails and banking details when you are online.
  
  
• Prime factorization leads to finding the highest common factor (HCF) and lowest common multiple (LCM) of larger numbers.
  
  
• A 'free gift' with this skill is that we can now also simplify fractions and ratios.
  
  
• This could help you if you want to work in pure mathematics, computing and internet security,

Preparing:

• Are we ready?
  Do you know the first ten prime numbers?
• How many more primes can you write down? Make a bid!
  Try to write them down before you reveal them using this widget:

• Before you start you ought to know:
  • what is meant by the key words: 'product', 'factors', 'multiples' and 'prime numbers'
  • your multiplication tables (times-tables) up to 10 × 10 pretty well
  • how to use indices to write small powers of a number like 23
  • divisibility tests to tell if any number is divisible by 2, 3, 4, 5, 6, 9 and 10
    
    
• Let's be sure you can explain: What makes a number prime?

• You will have a deeper understanding if you also know the rules of indices.

Discovering:

• Can you figure it out yourself from these examples?

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate using this tool:

Modeling:

• Here are some examples of people getting it right:

• Here are some examples of people getting it wrong in typical ways:

• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be?

• ... tell your learning partner. Convince them you're right.
  
  
• Explain how you know.
  
  
• How would you explain this to someone who was new to it?

Explaining:

• A lovely vizualisation is the Primitives tool by Alex McEachran:

• Another approach to explaining this might be mymaths: factor trees and HCF and mymaths: numbers & powers revision
• It doesn't work when the starting number is not an integer greater than or equal to 2
• An exception is when you start with a prime, in which case the tree is quite short.
• Watch out for factors other than 2: many pupils seem to think that you are only allowed to divide by 2 at each step
• A common mistake is to find pairs that sum to the number above instead of multiply to make it.
• Also watch out for dividing by 1, which doesn't really help much.
• You can check your result by using the Wolfram widget above
• In an exam, check by multiplying your answer back out to check you get the number you started with.
• We can prove this works: see Wikipedia: the fundamental theorem of arithemtic and Wolfram mathworld: the fundamental theorem of arithmetic

Practising:

• Have a heap of fun practising with the Manga High Sigma Prime game:

• practice with the virtual factor tree from the National Library of Digital Manipulatives
• you can record your success if you log in to Kahn Academy on a computer and try their prime factorization quiz or you could get your teacher to project a copy and do it as a class using multiple choice cards.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Explore the materials at MathIlluminated unit 1: the primes.
  • mathilluminated video
  • transcript of the video:
    

    MathIlluminated_01_trans.pdf
    

    • Details
    • Download
    • 145 KB
  • mathilluminated figurate numbers interactive tool
  • participants' guide:
    

    MathIlluminated_01_par.pdf
    

    • Details
    • Download
    • 433 KB
  • textbook chapter:
    

    MathIlluminated_01_txt.pdf
    

    • Details
    • Download
    • 1 MB
  • facilitators' guide:
    

    MathIlluminated_01_fac.pdf
    

    • Details
    • Download
    • 847 KB
    
    
    
• Research how public key encryption works. Make a poster to explain how the RSA algorithm uses large prime numbers to make a code.
• Research about different types of prime number such as Mersenne primes.
• Explore what is meant by the Fundamental Theorem of Arithmetic:

• This leads to...
• Now try...